TY - JOUR
T1 - Climbing up a random subgraph of the hypercube
AU - Anastos, Michael
AU - Diskin, Sahar
AU - Elboim, Dor
AU - Krivelevich, Michael
N1 - Publisher Copyright:
© 2024, Institute of Mathematical Statistics. All rights reserved.
PY - 2024
Y1 - 2024
N2 - Let Qd be the d-dimensional binary hypercube. We say that P = {v1, …, vk } is an increasing path of length k − 1 in Qd, if for every i ∈ [k − 1] the edge vivi+1 is obtained by switching some zero coordinate in vi to a one coordinate in vi+1. Form a random subgraph Qdp by retaining each edge in E(Qd) independently with probability p. We show that there is a phase transition with respect to the length of a longest increasing path around p =ed. Letαbe a constant and letp=αd. When 0 < α < e, then there exists a δ ∈ (0, 1) such that whp a longest increasing path in Qdp is of length at most (1 − δ)d. On the other hand, when α > e, whp there is a path of length d − 2 in Qdp, and in fact, whether it has length d − 2, d − 1, or d depends on whether the vertices (0, …, 0) and (1, …, 1) are in the giant connected component.
AB - Let Qd be the d-dimensional binary hypercube. We say that P = {v1, …, vk } is an increasing path of length k − 1 in Qd, if for every i ∈ [k − 1] the edge vivi+1 is obtained by switching some zero coordinate in vi to a one coordinate in vi+1. Form a random subgraph Qdp by retaining each edge in E(Qd) independently with probability p. We show that there is a phase transition with respect to the length of a longest increasing path around p =ed. Letαbe a constant and letp=αd. When 0 < α < e, then there exists a δ ∈ (0, 1) such that whp a longest increasing path in Qdp is of length at most (1 − δ)d. On the other hand, when α > e, whp there is a path of length d − 2 in Qdp, and in fact, whether it has length d − 2, d − 1, or d depends on whether the vertices (0, …, 0) and (1, …, 1) are in the giant connected component.
KW - bond percolation
KW - hypercube
KW - phase transition
UR - http://www.scopus.com/inward/record.url?scp=85211132659&partnerID=8YFLogxK
U2 - 10.1214/24-ECP639
DO - 10.1214/24-ECP639
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AN - SCOPUS:85211132659
SN - 1083-589X
VL - 29
JO - Electronic Communications in Probability
JF - Electronic Communications in Probability
M1 - 70
ER -